As shown in [1]}, we can solve this system explicitly:
\(C_k(t) &= Z(t)\left[C_k(0) - \int _0^tZ^{-1}(u)(\rho \,\mathrm {d}W_k(u) - \gamma \sigma _S\rho \,\mathrm {d}u)\right];\\Z(t) &= \exp \left[-(\mu + \gamma g_S)t - \gamma \sigma _S\,W_S(t) - \frac{1}{2}\gamma ^2\sigma _S^2t\right].\) 
